Disentangling topological degeneracy in the entanglement spectrum of one-dimensional symmetry protected topological phases

Abstract

One-dimensional valence bond solid (VBS) states represent the simplest symmetry protected topological phases. We show that their ground state entanglement spectrum contains both topological and non-topological structures. For the $SO(3)$ symmetric VBS states with odd-integer spins, the two-fold topological degeneracy is associated with an underlying $Z_{2}\times Z_{2}$ symmetry that protects the corresponding topological phase. In general, for the $SO(2S+1)$ symmetric VBS states with integer spins $S$, the corresponding protecting symmetry is identified as the $(Z_{2}\times Z_{2})^{S}$ symmetry, yielding the $2^{S}$-fold topological degeneracy. The topological degeneracy and associated protecting symmetry can be identified by a non-local unitary transformation, which changes the topological order of the VBS states into conventional ferromagnetic order.